Gödel's theorem may be demonstrated using arguments having an information-theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.

An attempt is made to apply information-theoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms. 2 G. J. Chaitin Key Words and Phrases: complexity of sets, computational complexity, difficulty of theoremproving, entropy of sets, formal systems, Godel's incompleteness theorem, halting problem, information content of sets, information content of axioms, information theory, information time trade-offs, metamathematics, random strings, recursive functions, recursively enumerable sets, size of proofs, universal computers CR Categories: 5.21, 5.25, 5.27, 5.6 1. Introduct...

We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and k-tuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and k-tuples of primes separated by fixed gaps. For example, if d is even, we define P d (x) = |{0 < n < x : n, n + d are both prime}|. They conjecture both that lim x## P d (x) P 2 (x) = # odd p|d p - 1 p - 2 and that P 2 (x) is asymptotic to 2 # p>2 # 1 - 1 (p - 1) 2 # # x 2 dy (log y) 2 . We will refer to the first equation as the "relative conjecture" and the second as the "absolute conjecture." There has been much numerical verification of these conjectures, and many attempts at proofs. Balog [1] proves a result that implies that the conjectures are true "on average," where the average is taken over the possible shapes of the k-tuples. Golubev [6] compares these conjectures with provable analogous limit results for patterns of numbers prime to n. Turan [18] relates such theorems to zeroes of the #-function, using the large sieve rather than Hardy and Littlewood's circle method. There are also many generalizations to specific fields. Most of those generalizations use "Conjecture H" of Sierpinski and Schinzel [14,15]. For example, Sierpinski [17] shows that Conjecture H implies the existence of infinitely many prime Gaussian integers di#ering by 2. Bateman and Horn [2,3] quote a quantitative form of Conjecture H which allows them to estimate the density of rational twin primes. Shanks [16] numerically verifies that the density of prime pairs of the form a + i, a + 2 + i in the Gaussian integers matches that of the quantitative form of Conjecture H. Rieger ...

I develop tools to amplify our mental senses: our intuition and reasoning abilities. The first five chapters -- based on the Order of Magnitude Physics class taught at Caltech by Peter Goldreich and Sterl Phinney -- form part of a textbook on dimensional analysis, approximation, and physical reasoning. The text is a resource of intuitions, problem-solving methods, and physical interpretations. By avoiding mathematical complexity, order-of-magnitude techniques increase our physical understanding, and allow us to study otherwise difficult or intractable problems. The textbook covers: (1) simple estimations, (2) dimensional analysis, (3) mechanical properties of materials, (4) thermal properties of materials, and (5) water waves. As an extended example of order-of-magnitude methods, I construct an analytic model for the flash sensitivity of a retinal rod. This model extends the flash-response model of Lamb and Pugh with an approximate model for steady-state response as a function of backgrou...

I develop tools to amplify our mental senses: our intuition and reasoning abilities. The first five chapters---based on the Order of Magnitude Physics class taught at Caltech by Peter Goldreich and Sterl Phinney---form part of a textbook on dimensional analysis, approximation, and physical reasoning. The text is a resource of intuitions, problem-solving methods, and physical interpretations. By avoiding mathematical complexity, order-of-magnitude techniques increase our physical understanding, and allow us to study otherwise di#cult or intractable problems. The textbook covers: (1) simple estimations, (2) dimensional analysis, (3) mechanical properties of materials, (4) thermal properties of materials, and (5) water waves. As an extended example of order-of-magnitude methods, I construct an analytic model for the flash sensitivity of a retinal rod. This model extends the flash-response model of Lamb and Pugh with an approximate model for steady-state response as a function of backgrou...

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Abstract. We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo m. Using an inclusion/exclusion argument and a cut-off of an infinite product suggested by Pólya, we obtain a heuristic formula for the “probability ” that a pair of consecutive prime numbers of size approximately x will be congruent to (a, a+d) modulo m. We demonstrate some symmetries of our formula. We test our formula and some of its consequences against data for x in various ranges. 1.